GRADING AND SCREENING 239 screen work of a more or less heavily loaded shaking screen, will be touched upon later. In screening with a single layer of particles on a flat screen the mere change of direction to which the particles are subjected by impacts does not modify the theory but helps it. If the aperture be square and with a side of 1 in. and n be made 8, 4, 2 and 1, the following tabulation can be made of the chance of grains 3^, Ji, J£ and 1 in. falling through the inch-square aperture and the probable number of squares such grains will have to cross before falling through an aperture of this size. SME OF GRAIN, DIAMETER, CHANCE OF FALLING IN SQUARE PROBABLE SQUARES, I-IN. SIDE, INCHES OF IN. SIDE TO CROSS, PROPORTIONAL TO 0 Infinity 0 H 3.27 0.31 Y± 1.29 0.78 Vz 0.33 3.00 1 0 Infinity The number of squares to cross is taken under the simplest possible assumption, viz., that the probability of a grain falling through an aperture begins de novo at each opening and that the grains do not follow any regular path. Capacity of Shaking Screens under the Chance Law.—If for different vai I, l/n be plotted as y and the chance as x, a series of conf ocal hyperbolas result and the radius of curvature will increase as the value of I is increaseu. The average value of the chance for each curve, the average chance, can be obtained by the quadrature of the curve between the limits y equals 0 and I and x equals infinity and 0 and dividing the result by I. There can be no quadrature of such an area but an approximate expression can be obtained by considering the curves as equilateral hyperbolas, taking the measure for I in a unit sufficiently small so as substantially to produce curves of this kind, and then considering the curves as tangent to the axis at distances I and to equal the quarter arc of a circle of this diameter. A more exact mode of solution would be to take l/n sufficiently small for the first term plotted so that while the position of x would be at a comparatively great distance from the origin it would be at a finite distance. If this be done the value of the chance can be determined by the methods of the integral calculus. When the average chance for different values of I is calculated it is seen at once that the average chance of grains going through the apertures of screens is directly proportional to the size of the aperture. Stated in a practical way, in a battery of flat shaking screens other factors such as size, shape, number of shakes per minute, etc., being equal, the capacity of the screen is proportional to the size of the opening.1 The bigger the opening the greater the capacity and the smaller the opening the less the capacity. Screening en masse on flat screens or screening with screens of other types the chance law is still the governing one. The gist of the argument on the chance law will be seen on reflecting that the more near a article is to the size of the aperture the more difficult it will be for it to pass through the screen opening and the more squares it will have to cross to be eliminated as a undersize grain. Second, since the material fed on the screens of a battery of screens consists of grains ranging in diameter from zero to a diameter but little greater than the width of the apertures, the limiting upper diameter of grain on each screen being not greater than the width of aperture of the next coarser one of the battery the ratio of the average diameter to the size of the opening becomes greater as 1 See p. 246, et seq.